Problem: Which of the cones below can be formed from a $252^{\circ}$ sector of a circle of radius 10 by aligning the two straight sides?

[asy]
draw((5.8,8.1)..(-10,0)--(0,0)--(3.1,-9.5)..cycle);
label("10",(-5,0),S);
label("$252^{\circ}$",(0,0),NE);
[/asy]

A. base radius = 6, slant =10

B. base radius = 6, height =10

C. base radius = 7, slant =10

D. base radius = 7, height =10

E. base radius = 8, slant = 10
Answer: The slant height of the cone is equal to the radius of the sector, or $10$.  The circumference of the base of the cone is equal to the length of the sector's arc, or $\frac{252^\circ}{360^\circ}(20\pi) = 14\pi$.  The radius of a circle with circumference $14\pi$ is $7$.  Hence the answer is $\boxed{C}$.